Question

How do you evaluate the integral `int xe^(-x^2)dx` from `-oo` to `oo`?

Answer 1

`int_-∞^∞xe^(-x^2)=0`

Explanation:

Let `color(red)(u=e^(-x^2))`

`du=-2xe^(-x^2)dx`

`color(blue)(-(du)/2=xe^(-x^2)dx)`

So
`intxe^(-x^2)dx`

`intcolor(blue)(-(du)/2)`

`=-color(red)(u)/2+C` where C is a constant

`=-color(red)(e^(-x^2))/2+C`

Therefore,

`int_-∞^∞xe^(-x^2)=(-e^(-(-∞)^2)/2)-(-e^(-∞^2)/2)`
`int_-∞^∞xe^(-x^2)=(-e^(-∞)/2)-(-e^(-∞)/2)`

Knowing that `e^(-∞)=0`

Then,

`int_-∞^∞xe^(-x^2)=0-0=0`